| Abstract Convexity and Global Optimization 2000 Edition Contributor(s): Rubinov, Alexander M. (Author) |
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ISBN: 079236323X ISBN-13: 9780792363231 Publisher: Springer
Binding Type: Hardcover - See All Available Formats & Editions Published: May 2000 Annotation: This book consists of two parts. Firstly, the main notions of abstract convexity and their applications in the study of some classes of functions and sets are presented. Secondly, both theoretical and numerical aspects of global optimization based on abstract convexity are examined. Most of the book does not require knowledge of advanced mathematics. Classical methods of nonconvex mathematical programming, being based on a local approximation, cannot be used to examine and solve many problems of global optimization, and so there is a clear need to develop special global tools for solving these problems. Some of these tools are based on abstract convexity, that is, on the representation of a function of a rather complicated nature as the upper envelope of a set of fairly simple functions. Audience: The book will be of interest to specialists in global optimization, mathematical programming, and convex analysis, as well as engineers using mathematical tools and optimization techniques and specialists in mathematical modelling. Click for more in this series: Nonconvex Optimization and Its Applications |
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| Additional Information |
| BISAC Categories: - Mathematics | Game Theory - Mathematics | Applied - Medical |
| Dewey: 519.3 |
| LCCN: 00039102 |
| Series: Nonconvex Optimization and Its Applications |
| Physical Information: 1.13" H x 6.14" W x 9.21" L (1.97 lbs) 493 pages |
| Features: Bibliography, Index |
| Descriptions, Reviews, Etc. |
| Publisher Description: Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac- complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu- ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh- bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func- tion. Since f(y) h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support. |
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